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\title{Octonion matrix algebras}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
``New Trends in Quaternions and Octonions''
\\
Universidade de Tr\'as-os-Montes e Alto Douro
\\
November 26, 2021
}
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{\small
\begin{center}
Based on joint work with Arezoo Zohrabi
\\
J. Nonlin. Math. Phys. 28 (2021), 108--122
\end{center}
}
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\begin{frame}{Octonion matrix algebras}
$S^+(M_n(\mathbb O), J) = \set{A \in M_n(\mathbb O)}{J(A) = A}$ \hskip 10pt
Hermitian
\bigskip
$S^-(M_n(\mathbb O), J) = \set{A \in M_n(\mathbb O)}{J(A) = -A}$ \hskip 3pt
skew-Hermitian
\bigskip
$J: (a_{ij}) \mapsto (\overline{a_{ji}})$ an involution
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\begin{frame}{Why bother?}
$\boldsymbol{S^+(M_n(\mathbb O), J)}$:
\smallskip
$n=1$: the ground field $K$
\smallskip
$n=2$: the $10$-dimensional simple Jordan algebra of symmetric \\
\hskip 34pt bilinear form
\smallskip
$n=3$: the famous $27$-dimensional exceptional simple Jordan \\
\hskip 34pt algebra
$n\ge 4$: no longer Jordan, but appear in $M$-theory
\bigskip
$\boldsymbol{S^-(M_n(\mathbb O), J)}$:
\smallskip
$n=1$: the $7$-dimensional simple Malcev algebra $\mathbb O^{-}$.
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\begin{frame}{Some references}
\begin{itemize}
\item
M. Bremner and I. Hentzel,
\emph{Identities for algebras of matrices over the octonions},
J. Algebra \textbf{277} (2004), no.1, 73--95.
\item
P. Jordan, \emph{Zur Theorie nicht-assoziativer Algebren},
Akad. Wiss. Lit. Mainz Abh. Math.-Natur. Kl. 1968, no.2, 27--38.
\item J. Lukierski and F. Toppan,
\emph{Generalized space-time supersymmetries, division algebras and octonionic M-theory},
Phys. Lett. B \textbf{539} (2002), no.3-4, 266--276.
\item H. Petyt, \emph{Derivations of octonion matrix algebras},
Comm. Algebra \textbf{47} (2019), no.10, 4216--4223.
\item
H. R\"uhaak,
\emph{Matrix-Algebren \"uber einer nicht-ausgearteten Cayley-Algebra},
PhD Thesis, Univ. Hamburg, 1968.
\end{itemize}
\end{frame}
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\begin{frame}{Simplicity}
\begin{block}{Theorem 1}
The algebras $S^+(M_n(\mathbb O), J)$ and $S^-(M_n(\mathbb O), J)$ are simple.
\end{block}
\bigskip
Method of the proof: use realizations
$$
S^{\pm}(M_n(\mathbb O), J) \simeq
M_n^{\pm}(K) \otimes 1 + M_n^{\mp}(K) \otimes \mathbb O^{-} ,
$$
and a variant of the Jacobson density theorem.
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\begin{frame}{$\delta$-derivations and associative forms}
$$
D(xy) = \delta D(x)y + \delta xD(y)
$$
\bigskip
\begin{block}{Theorem 2}
$\delta$-derivations of $S^+(M_n(\mathbb O), J)$ and $S^-(M_n(\mathbb O), J)$
are trivial (i.e., either the usual derivations, or multiplications by a
scalar).
\end{block}
\bigskip
(Earlier derivations were computed by Petyt).
\bigskip
\begin{block}{Theorem 3}
Symmetric associative forms on $S^+(M_n(\mathbb O), J)$ and
$S^-(M_n(\mathbb O), J)$ are:
$$
(X, Y) \mapsto \Tr\,(XY + \overline X \, \overline Y) .
$$
\end{block}
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\begin{frame}{Further questions}
\begin{itemize}
\item Automorphisms? Conjecture: $G_2 \times SO(n)$.
\item Identities?
\item Subalgebras?
\item Deformations?
\end{itemize}
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{\Huge That's all. Thank you.}
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